Algebra2D

Complex Numbers - Theory & Concepts - De Moivre Theorem

Understanding imaginary numbers, complex operations, the complex plane, and De Moivre's Theorem.

Complex Number z

z = 1.20 \left(\cos 45^\circ + i \sin 45^\circ\right)

z = 1.20 e^{i 0.785}

Magnitude r (Radius)1.20

Angle θ (Degrees)45°

Power nn = 3

Result:

z^{n}

z^{3} = 1.20^{3} \left(\cos (3 \cdot 45^\circ) + i \sin (3 \cdot 45^\circ)\right)

z^{3} = 1.728 \left(\cos 135^\circ + i \sin 135^\circ\right)

Notice that the magnitude raises geometrically to 1.728, while the angle multiplies linearly to 135° (or 135° coterminal).

z (Original)

z^{n}

(Result)

Intermediates

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